Hierarchical Scaled (VAT) Description of Thermal Transport in Heterogeneous (including Porous) Media.
When heterogeneous medium recognized as being in dependence on lower (smaller) scale physical phenomena- Bottom-Up communication, physical properties connection described, or vise versa - the Top-Down mechanisms or processes prevail as, for example, usually the heat flux being applied to the top scale piece of medium. Then its physical and mathematical descriptions on the Upper Scale (usually scale of thermal transport phenomena of global interest) need to be considered in a way incorporating the influence (dependence) of lower scale energy transport descriptions and mathematical modeling onto the Upper scale and vice versa. Most often arising the situation when the two levels of hierarchy and two - or three phases in media allow connecting phenomena in the neighboring scales. This theory, among methods in use today, has special attributes that enable explanations of hierarchical phenomena to be obtained because it is able to combine lower (first) scale phenomena, with the Upper Heterogeneous medium scale phenomena, and interfacial and morphology features simultaneously with mathematical rigor. As long as the energy and heat transport in various scales having homogeneous different mechanisms, it is important to preserve those mechanisms while connecting different scale models through the VAT.
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Why is it Different from Homogeneous Thermal Physics in Heterogeneous Media Description?
The main differences are due to different governing equations for the UPPER scale physics description. Those are mainly because the Heterogeneous Whitaker-Slattery-Anderson-Marle (WSAM) kind of theorems explains why those equations are different.
Fundamentals of Scaling Heterogeneous Science

What is the VAT Thermal Physics in Heterogeneous Media?
The VAT description of heat and other energy fields transport in heterogeneous medium includes the mathematical models, governing equations for both scales (at least) of the medium - lower spatial scale homogeneous governing equations and the upper scale heterogeneous governing equations. It is worth to note here that for few structures - morphologies, as for the capillary and globular (with spherical inclusions) morphologies of porous and composite media some exact solutions (first in the business) were obtained. For example, for capillary morphology - ExactClosure-IMECE-2001-24261 ( 589k) . Additional morphologies solutions see this and other sections of this website.

Are there any other Methods and Theories available?
There were many theories and methods devoted to heterogeneous media thermal physics. Numerous books and textbooks are based on bulk one scale consideration approximate methods. That is the conventional (and wrong, obsolete for Heterogeneous, porous, scaled, and hierarchical media) kind of analysis and modeling. See more explanations in -
Fundamentals of Scaling Heterogeneous Science